Constant-Time Algorithms for Constrained Triangulations on Reconfigurable Meshes

A number of applications in computer-aided manufacturing, CAD, and computer-aided geometric design ask for triangulating pieces of material with defects. These tasks are known collectively as constrained triangulations. Recently, a powerful architecture called the reconng-urable mesh has been proposed: in essence, a reconngurable mesh consists of a mesh-connected architecture augmented by a dynamically reconngurable bus system. The main contribution of this paper is to show that the exibility of the reconngurable mesh can be exploited for the purpose of obtaining constant-time algorithms for a number of constrained triangulation problems. These include triangulating a convex planar region containing any constant number of convex holes, triangulating a convex planar region in the presence of a collection of rectangular holes, and triangulating a set of ordered line segments. Speciically, with a collection of O(n) such objects as input, our algorithms run in O(1) time on a reconngurable mesh of size n n. To the best of our knowledge, this is the rst time constant time solutions to constrained triangulations are reported on this architecture.